The exchange frequency in crystal ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is calculated from first principles with a combination of path-integral Monte Carlo method and a method used in classical statistical mechanics to determine free-energy differences. The frequency of nearest-neighbor exchange at melting density is 0.46 mK, that of triple exchange is 0.19 mK, and that of four-particle planar exchange is 0.27 mK. These exchange frequencies are within 30% of the values obtained from the empirical multiple-exchange model and agree with measurements. Many other types of high-order cyclic exchanges make significant contributions to thermodynamic properties, showing that ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is more complex than previously thought.

• Received 21 July 1986

DOI:https://doi.org/10.1103/PhysRevLett.58.1648